Download Modeling and Simulation of the Immune System as a Self

C H A P T E R
F O U R
Modeling and Simulation of the
Immune System as a Self-Regulating
Network
Peter S. Kim,* Doron Levy,† and Peter P. Lee‡
Contents
1. Introduction
1.1. Complexity of immune regulation
1.2. Self/nonself discrimination as a regulatory phenomenon
2. Mathematical Modeling of the Immune Network
2.1. Ordinary differential equations
2.2. Delay differential equations
2.3. Partial differential equations
2.4. Agent-based models
2.5. Stochastic differential equations
2.6. Which modeling approach is appropriate?
3. Two Examples of Models to Understand T Cell Regulation
3.1. Intracellular regulation: The T cell program
3.2. Intercellular regulation: iTreg-based negative feedback
4. How to Implement Mathematical Models in Computer Simulations
4.1. Simulation of the T cell program
4.2. Simulation of the iTreg model
5. Concluding Remarks
Acknowledgments
References
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Abstract
Numerous aspects of the immune system operate on the basis of complex
regulatory networks that are amenable to mathematical and computational
modeling. Several modeling frameworks have recently been applied to simulating the immune system, including systems of ordinary differential equations,
delay differential equations, partial differential equations, agent-based models,
*
{
{
Department of Mathematics, University of Utah, Salt Lake City, Utah, USA
Department of Mathematics and Center for Scientific Computation and Mathematical Modeling
(CSCAMM), University of Maryland, College Park, Maryland, USA
Division of Hematology, Department of Medicine, Stanford University, Stanford, California, USA
Methods in Enzymology, Volume 467
ISSN 0076-6879, DOI: 10.1016/S0076-6879(09)67004-X
#
2009 Elsevier Inc.
All rights reserved.
79
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Peter S. Kim et al.
and stochastic differential equations. In this chapter, we summarize several
recent examples of work that has been done in immune modeling and discuss
two specific examples of models based on DDEs that can be used to understand
the dynamics of T cell regulation.
1. Introduction
The immune system plays a vital role in human health, with more than
15% of genes in the human genome being linked to immune function
(Hackett et al., 2007). The immune system is generally thought to protect
against external invaders, such as bacteria, viruses, and other pathogens,
while ignoring self. The mechanisms by which the immune system discriminates between self and nonself are becoming elucidated, but are far
from being completely understood. Lymphocytes (T and B cells) express
antigen receptors generated via novel combinations of gene (V, D, J)
segments. This creates an extraordinarily diverse repertoire of unique antigen receptors (>107 T cell receptors, TCR, in T cells (Arstila et al., 1999);
>108 immunoglobulins, Ig, in B cells (Rajewsky, 1996)) that can respond to
potentially all pathogens. However, self-reactive lymphocytes that are also
generated in the process could cause autoimmunity if left unchecked.
Newly generated T cells mature within the thymus: 95% die during this
process, due to strong binding to self antigens (negative selection) or lack of
sufficient signaling (positive selection). Thymic selection is a powerful force
that shapes the mature T cell repertoire; this process is referred to as central
tolerance. It is now known that potentially autoreactive T cells still persist
after thymic selection, so other mechanisms must be operative to keep these
in check to maintain peripheral tolerance. A major area of focus in immunology in recent years is regulatory T cells (Tregs), which suppress other
immune cells and play an important role in peripheral self tolerance. While
there is no organ equivalent to the thymus for B cells, two early tolerance
checkpoints regulate developing autoreactive human B cells: the first one at
the immature B cell stage in the bone marrow, and the second one at the
transition from new emigrant to mature naive B cells in the periphery
(Meffre and Wardemann, 2008). As the thymus involutes by young adulthood, how potentially autoreactive T cells are deleted from then on is
unclear. New experimental and modeling work suggest that the thymus
may play a less prominent role than generally thought in the development of
the peripheral T cell pool, even in persons below age 20 (Bains et al., 2009).
While the self/nonself view of immunology makes sense and holds true
by and large, exceptions exist upon closer inspection. Since cancer cells are
of self origin, it was assumed for decades that the immune system ignores
cancer. Yet, approximately 80% of human tumors are infiltrated by T cells,
Modeling and Simulation of the Immune System
81
which appear to have beneficial effects (Galon et al., 2006; Nelson, 2008).
Tumor-infiltrating lymphocytes (TILs) have been expanded in vitro and
their targets have been identified. Contrary to initial expectations, most
tumor-infiltrating T cells were found to be directed against self, nonmutated
antigens. Such antigens are commonly referred to as tumor-associated
antigens or TAAs. Many of the TAAs identified thus far have been in the
setting of melanoma (Kawakami and Rosenberg, 1997; Rosenberg,
2001)—the most common ones include MART (melanoma antigen recognized by T cells), gp100, and tyrosinase; others include MAGE, BAGE,
GAGE, and NYESO. TAAs have also been identified for breast cancer
(e.g., HER-2/neu (Sotiropoulou et al., 2003), MUC (Böhm et al., 1998)),
leukemia (e.g., proteinase 3 (Molldrem et al., 1999), WT1 (Oka et al.,
2000)), and colon cancer (e.g., CEA (Fong et al., 2001)). Hence, tumor
immunity is a form of autoimmunity (Pardoll, 1999). How TAAs which are
self, nonmutated proteins break tolerance in the setting of cancer remains
poorly understood. This adds complexity to the puzzle of immune
regulation.
During a typical infection, the immune response unfolds in multiple
waves. The cascade begins with almost immediate responses by innate
immune cells, such as neutrophils, which create an inflammatory microenvironment that subsequently attracts dendritic cells and lymphocytes to
initiate the adaptive immune response. Perhaps one reason the immune
system operates in a series of successive waves rather than in one continuous,
concentrated surge is that each burst of immune cells has to be tightly
regulated, since some primed cells could potentially give rise to an uncontrolled autoimmune response. Most immune cells exist in different states
(resting/active, immature/mature, naı̈ve/effector/memory), which provide
additional regulatory mechanisms. What controls the magnitude and duration of each individual response, how does one response give way to
another, and induce cellular state changes? More generally, how does
the immune system work as such a multifaceted, yet robustly controlled
network? In this chapter, we show how principles from mathematical
modeling can shed light into understanding how the immune system functions as a self-regulating network. The complexity of the immune system,
with emergent properties and nonlinear dynamics, makes it amenable to
computational methods for analysis.
1.1. Complexity of immune regulation
As knowledge of the immune system grows it becomes increasingly clear
that most immune ‘‘decisions’’ (e.g., whether to attack or tolerate a certain
target, or whether to magnify or suppress an immune response) are not
made autonomously by individual cells or even by a few isolated cells.
Instead, most immune responses result from a multitude of interactions
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among various types of cells, continually signaling to one another via cell
contact and cytokine-mediated mechanisms.
For example, various types of T cells interact to drive cytotoxic T cell
expansion and produce an overall immune response. To begin, T cells
must be activated in the lymph node by antigen-presenting cells (APCs),
primarily dendritic cells, that present stimulatory or suppressive signals
depending on what signals they received while interacting with other cells
and cytokines in the surrounding tissue. In the event of infection, APCs
usually start by stimulating CD4þ T cells, which begin to multiply and
secrete IL-2 and other growth signals that lead to increased activity in
the lymph node. Shortly afterward, the cytotoxic (CD8þ) T cells get
stimulated and begin to proliferate rapidly. Cytotoxic T cells also produce
a small amount of IL-2, but mostly direct their energy to extensive
proliferation.
Even then, T cell activation follows a more multifaceted route than that
already described, for upon stimulation, helper (CD4þ) T cells commit to
one of two maturation pathways, Th1 and Th2, depending on the type of
stimulation by APCs and cytokine signals. These pathways direct the
adaptive immune response toward cellular or humoral immunity, the first
of which is mediated by T cells and macrophages and the latter by B cells
and antibodies. Furthermore, in a coregulating network, these separate
responses serve to promote their own advancement while suppressing the
other. Specifically, activated Th1 cells release IFN-g, which promote Th1
differentiation while hindering Th2 production, and conversely, activated
Th2 cells release IL-4 and IL-10, which promote Th2 production while
hindering Th1 cells. Even from this simplified perspective, CD4þ T cell
differentiation is governed by a regulatory network, composed of two
negatively coupled positive feedback loops.
Another type of T cell associated with the CD4þ family is the regulatory
T cell (Treg). As far as currently known, these cells function as a global,
negative feedback mechanism that suppresses all activated T cells, downregulates the stimulatory capacity of APCs, and secretes immunosuppressive
cytokines. These cells either emerge directly from the thymus with regulatory capability and are called naturally occurring regulatory T cells
(nTregs), or differentiate from nonregulatory T cells after activation and
are called antigen-induced regulatory T cells (iTregs). The precise mechanisms governing Treg-mediated regulation are not well understood,
although clear evidence shows that Tregs play an essential role in maintaining self tolerance and immune homeostasis (Sakaguchi et al., 1995, 2008).
For example, Tregs influence the extent of memory T cell expansion
following an immune response via an IL-2-dependent mechanism
(Murakami et al., 1998). Furthermore, Tregs may control the extent
of effector T cell proliferation during an acute immune response via an
IL-2-dependent feedback mechanism (Sakaguchi et al., 2008).
Modeling and Simulation of the Immune System
83
Shifting to another aspect of the adaptive immune response, B cells can be
activated by Th2 cells as discussed above, but they can also respond to antigen
without T cell intervention. Many antigens, especially those with repeating
carbohydrate epitopes such as those that come from bacteria, can stimulate B
cells without T cell intervention. Furthermore macrophages can also display
repeated patterns of the same antigen in a way that instigates B cell activation.
Yet, most antigens are T cell-dependent, and B cells usually require T cell
interaction to achieve maximum stimulation. Nonetheless, not only do T
cell-independent mechanisms for B cell activation exist, but experimental
evidence shows that B cells also play a role in regulating T cell responses. In
particular, the balance between IgG and IgM antibodies secreted by B cells
directs the immune response either toward monocytic cells which favor Th1
production or toward further B cell activity which favor Th2 production
(Bheekha Escura et al., 1995). In later work, Casadevall and Pirofski propose
that IgM and IgG may even direct the course of the T cell response by
playing proinflammatory and anti-inflammatory roles (Casadevall and
Pirofski, 2003, 2006). Hence, T cell/B cell interactions are not unequivocally unidirectional, since stimulatory and suppressive mechanisms operate in
a feedback loop through which each cell subpopulation reciprocally
influences the other. Furthermore, B cells also exhibit a high level of
self-regulation, since antigen-specific antibody responses can be amplified
or reduced by several hundredfold via an antibody-mediated feedback
mechanism (Heyman, 2000, 2003).
Although this summary only touches a small part of possible immune
behavior, it is clear from the myriad interactions among diverse immune
cells that nearly all responses are regulated by a huge network of positive and
negative feedback loops that consistently keep the global system in check.
1.2. Self/nonself discrimination as a regulatory phenomenon
Another critical aspect of the immune system is self/nonself discrimination.
This term refers to the capacity of the immune system to decide whether a
particular target is a virulent pathogen, a harmless foreign body, or a normal
and healthy self cell. The ensuing immune response must adjust drastically
based on the verdict of this decision. Discrimination between target types is
largely antigen-based. Certain pathogens, such as bacteria, microbes, and
parasites, present protein and carbohydrate sequences that never appear on
normal tissue, conspicuously marking them as nonself. The immune system
must continually learn over time to recognize certain peptide sequences as
normal, while continuing to recognize other sequences as foreign.
Until recently, the prevailing view was that antigen recognition worked
by a lock and key mechanism in which adaptive immune cells expressed
specific antigen receptors that only responded to one or a few peptide
sequences, making it straightforward to see how the immune system
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could avoid autoimmunity by removing any immune cells that had a chance
of reacting with self antigen. The distinction between self and nonself
antigen became blurred, however, when experimental studies revealed
that the mature repertoire still maintains self-reactive immune cells; mice
depleted of naturally occurring Tregs invariably develop autoimmune disease (Sakaguchi et al., 1995). Furthermore, experimental and quantitative
results showed that a high level of cross-reactivity is a central feature of the
T cell repertoire (Mason, 1998). These results indicated that T cells react to
a range of peptide sequences and that a T cell that primarily reacts to foreign
antigen could also potentially cross-react with some self-antigen, thus giving
rise to an autoimmune, bystander response against healthy cells.
Due to the intrinsic cross-reactivity of antigen receptors and the inevitable presence of self-reactive immune cells, successful self/nonself discrimination cannot occur as the result of a simple black and white mechanism
operating at the individual immune cell level. Instead, this process must
emerge from a self-regulatory immune network. Along this line, a novel
view of self/nonself discrimination is emerging as a group phenomenon
resulting from interactions among several immune agents, including APCs,
effector T cells, Tregs, and their molecular signals (Kim et al., 2007).
2. Mathematical Modeling of the Immune
Network
As mentioned above, the immune system operates according to a
diverse, interconnected network of interactions, and the complexity of
the network makes it difficult to understand experimentally. On one
hand, in vitro experiments that examine a few or several cell types at a
time often provide useful information about isolated immune interactions.
However, these experiments also separate immune cells from the natural
context of a larger biological network, potentially leading to nonphysiological behavior. On the other hand, in vivo experiments observe phenomena
in a physiological context, but are usually incapable of resolving the contributions of individual regulatory components. To provide a particular
example of this shortcoming, our understanding of Treg-mediated regulation and its effect on the immune response is still very poor, even though the
majority of individual Treg interactions have already been thoroughly
described. This problem of connecting complex, global phenomena to
basic interactions extends over a wide range of immunological questions.
How, then, can we take individual-scale results that have been established and connect them to large-scale phenomena? This gap in immunological knowledge provides a fruitful ground for mathematical modeling
and computational science. In the following sections, we provide specific
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examples of what approaches from mathematical modeling have been
applied and what insights have been gained. Table 4.1 gives a summary of
advantages, disadvantages, and examples for each modeling approach.
2.1. Ordinary differential equations
Mathematical models based on systems of ordinary differential equations
(ODEs) are the most common as these types of models have been used for
cancer immunology (de Pillis et al., 2005; Moore and Li, 2004), natural
killer cell responses (Merrill, 1981), B cell responses (Lee et al., 2009; Shahaf
et al., 2005), B cell memory (De Boer and Perelson, 1990; De Boer et al.,
1990; Varela and Stewart, 1990; Weisbuch et al., 1990), Treg dynamics
(Burroughs et al., 2006; Carneiro et al., 2005; Fouchet and Regoes, 2008;
León et al., 2003, 2004, 2007a,b), and T cell responses (Antia et al., 2003;
Wodarz and Thomsen, 2005 to name a few examples.
The primary advantage of ODE modeling is that this model structure has
already been extensively applied in the study of reaction kinetics and other
physical phenomena. In addition, the mathematical analysis of these systems
is relatively simple compared to other types of models and their solutions
can be computationally simulated with great efficiency. That is to say, these
models can be made extremely complex, before becoming computationally
unfeasible.
For example, Merrill constructs an ODE model of NK cell dynamics
(Merrill, 1981). In his model, NK cells represent an immune surveillance
population that responds immediately to stimulation without the need of
prior activation or proliferation. Using this model, he discusses how the NK
population could trigger a subsequent T cell response, if necessary, by
releasing stimulatory cytokines, such as IFN-g.
In a model focusing on a different aspect of the immune network,
Fouchet and Regoes consider interactions between T cells and APCs to
explain self/nonself discrimination (Fouchet and Regoes, 2008). In their
model, precursor T cells differentiate into either effector or regulatory
T cells depending on whether the stimulation from the APC is immunogenic or tolerogenic. The differentiated effector and regulatory T cells then
turn around and drive other APCs to become immunogenic or tolerogenic
in two competing positive feedback loops. Furthermore, Tregs also suppress
effector T cells. Using this model, Fouchet and Regoes demonstrate how
this feedback network causes the immune response to commit to either a
fully immunogenic or a fully tolerogenic response, depending on the initial
concentration, growth rate, and strength of antigenic stimulus of the target.
They also consider how perturbations in the target population may lead to
switches between the two network states.
Modeling the adaptive immune system as a whole, Lee et al. construct
a comprehensive ODE model incorporating APCs, CD4þ T cells, CD8þ
Table 4.1 Advantages, disadvantages, and examples of each modeling approach: ODEs, DDEs, PDEs, SDEs, and ABMs
Modeling
approach
Advantages
Disadvantages
Examples
ODE
Computationally efficient,
describes complex
systems elegantly, simple
mathematical analysis
easy to formulate
Does not capture spatial
dynamics or stochastic
effects
DDE
Captures delayed feedback,
computationally
efficient
Captures spatial dynamics
and age-based behavior
Does not capture spatial
dynamics or stochastic
effects
Computationally
demanding, complex
mathematical analysis
Computationally
demanding, difficult to
analyze mathematically
Highly computationally
demanding, difficult to
analyze mathematically
de Pillis et al. (2005), Moore and Li (2004), Merrill
(1981), Lee et al. (2009), Shahaf et al. (2005),
De Boer et al. (1990), De Boer and Perelson
(1990), Varela and Stewart (1990), Weisbuch
et al. (1990), Burroughs et al. (2006), Carneiro
et al. (2005), Fouchet and Regoes (2008),
León et al. (2004), León et al. (2003),
León et al. (2007a,b)
Kim et al. (2007), Colijn and Mackey (2005)
PDE
SDE
Captures stochastic effects
ABM
Captures spatial dynamics
and individual diversity,
captures stochastic
effects, easy to formulate
Antia et al. (2003), Onsum and Rao (2007)
Figge (2009)
Catron et al. (2004), Scherer et al. (2006),
Figge et al. (2008), Casal et al. (2005)
Modeling and Simulation of the Immune System
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T cells, B cells, antibodies, and two immune environments, lungs and
lymph nodes (Lee et al., 2009). Using their model, Lee et al. investigate
multiple scenarios of infection by influenza A virus, and study the effects of
immune population levels, functionality of immune cells, and the duration
of infection on the overall immune response. They propose that antiviral
therapy reduces viral spread most effectively when administered within two
days of exposure. Their highly intricate, multifaceted model demonstrates
the ability of mathematical techniques to capture a multitude of dynamic
interactions over a broad spectrum of cell types.
2.2. Delay differential equations
Systems of ODEs are finite dimensional dynamical systems, while delay
differential equations (DDEs) and partial differential equations (PDEs) are
infinite-dimensional dynamical systems. As a result, DDEs and PDEs
require more computational and analytical complexity than their finite
dimensional counterparts. However, infinite-dimensional systems come
with unique modeling advantages.
In general, DDEs are simpler than PDEs. DDE models are also similar in
structure to ODE models, except that they explicitly include time delays.
Many biological processes exhibit delayed responses to stimuli, and DDE
models allow us to understand the effects of these delays on a feedback
network.
An example of a model that makes use of DDEs is the work by Colijn
and Mackey (2005) in which they model the development of neutrophils
from stem cells (i.e., neutrophil hematopoiesis). Neutrophils that have
attained maturity release a molecular signal that causes cells earlier in
development to stop differentiating. Ideally, this signaling gives rise to a
delayed negative feedback that ultimately stabilizes the neutrophil population at an equilibrium. However, the long delay in the signal permits a
situation in which the neutrophil population never stabilizes, but continues
to oscillate from unusually high to unusually low levels. Colijn and Mackey
connect this oscillatory dynamic to cyclical neutropenia, a disease that
causes patients to have periodically low levels of neutrophils.
Another example is our recent work in which we devise a mathematical
model to study the regulation of the T cell response by naturally occurring
Tregs (Kim et al., 2007). In this model, we consider a variety of immune
agents, including APCs, CD4þ T cells, CD8þ T cells, Tregs, target cells,
antigen, and positive and negative growth signals. Furthermore, each
immune population can migrate between two distinct environments: the
lymph node and the tissue. We also consider various time scales, such as a
long delay between initial CD8þ T cell stimulation and full activation and a
much shorter delay for each T cell division. The delays cause the CD8þ
response to initiate with a time lag after the CD4þ response. In addition,
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the delay due to cell division ensures that the Treg response develops more
slowly than the other two T cell responses, allowing a small time window of
unrestricted T cell expansion.
The delays produce another unexpected phenomenon, a two-phase
cycle of T cell maturation. In the first phase, CD4þ T cells expand and
secrete positive growth signal allowing CD8þ T cells to proliferate rapidly,
whereas in the second phase, the Treg population catches up to the original
effector T cell population and begins suppressing T cell activity, causing a
sudden shift from proliferation to emigration from the lymph node into the
peripheral tissue, where CD8þ T cells can more effectively eliminate the
target population.
From a practical point of view, DDE models are only slightly more
complex than ODE models to simulate numerically. Evaluating DDE
systems reduces to recording the past history of all populations throughout
the simulation. Hence, with only a slight increase in computational complexity, DDE models widely expand the repertoire of phenomena that can
be attained.
2.3. Partial differential equations
PDE models capture more complexity than DDE and ODE models. In
biological modeling, PDEs are often applied in two ways, age-structured
and spatio-temporal models.
Age-structured models account for the progression of individual cells or
members through a scheduled development process. As many organisms
exhibit behaviors that depend on their maturity and developmental level,
age-structured models provide a useful framework for modeling internal
development of an organism over time.
For example, Antia et al. formulate an age-structured model to simulate
the progression of cytotoxic T cells through an autonomous T cell proliferation program (Antia et al., 2003). According to the program, activated
T cells enter into a scheduled period of expansion, and then relative
stabilization, followed by a period of contraction, and then restabilization
at a lower level. These four stages of scheduled development comprise the T
cell proliferation program. Using this model, they study the effect of variations in the T cell program on the level and duration of cytotoxic T cell
responses. Furthermore, they conclude that T cell responses that are governed by autonomous, intracellular programs will execute similarly despite a
wide range of antigen stimulation levels. This latter phenomenon has also
been observed experimentally (Kaech and Ahmed, 2001; Mercado et al.,
2000; van Stipdonk et al., 2003).
Returning to the notion of regulation, a T cell program such as the one
modeled by Antia et al. (2003), or any other scheduled developmental
process, implies a system of internal self-regulation that may be invisible
Modeling and Simulation of the Immune System
89
to the external network, but that results from diverse interactions within the
cell. Due to the inherent difficulty of simultaneously modeling feedback
networks on intracellular and extracellular levels, age-structured models
provide an efficient tool for investigating the interactions between internal
and external regulatory mechanisms.
Another classical and highly useful application of PDE models is modeling spatio-temporal dynamics. Using this approach, Onsum and Rao
develop a PDE model for neutrophil migration toward a site of infection
by moving toward higher chemical concentrations (Onsum and Rao,
2007). They simulate how two chemical signals interacting in an antagonizing manner allow neutrophils to orient themselves within the chemical
gradient. Their PDE model is composed of a system of diffusion and
chemotaxis equations in one space dimension.
From the viewpoint of deterministic, differential equations, PDEs provide the most powerful mathematical modeling tool that captures the
broadest range of biological phenomena. These models have, however,
the potential to be significantly more computationally demanding than
ODE and DDE systems.
2.4. Agent-based models
The concept of an agent-based model (ABM) refers to a different modeling
philosophy than that used in differential equation systems. First of all, ABMs
deal with discrete and distinguishable agents, such as individual cells or
isolated molecules, unlike differential equations, which deal with collective
populations, such as densities of cells. In addition, ABMs easily allow us to
account for probabilistic uncertainty, or stochasticity, in biological interactions. For example, in a stochastic ABM, an individual agent only changes
state or location at a certain probability and not by following a deterministic
process. Finally, as with PDEs, most ABMs consider the motion of agents
through space.
A powerful application of ABMs is demonstrated by Catron et al. (2004).
They devise a sophisticated ABM to simulate the interaction between a
T cell and a dendritic cell in the lymph node. By observing repeated
simulations of T cell–DC interactions, they obtain estimates of the frequency of T cell–DC interactions and the expected time for T cells to
become fully stimulated.
In another ABM, Scherer et al. simulate T cell competition for access to
binding sites on mature antigen-bearing APCs (Scherer et al., 2006). As in
standard first-order reaction kinetics (i.e., the law of mass action), T cells
interact with APCs with a probability proportional to the product of their
two populations. Furthermore, each APC possesses a finite number of
antigen binding sites that can each present either of the two types of antigen
simulated in the model. Using their model, Scherer et al. determine that the
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nature of T cell competition changes depending on the level of antigen
expressed by the APCs. More specifically, under low antigen expression,
T cells of the same antigen-specificity are more likely to compete, allowing
for the coexistence of multiple T cell responses against different target
epitopes. On the other hand, under high antigen expression, T cell competition becomes more indiscriminate, ultimately allowing highly reactive
T cell populations to competitively exclude T cell populations that are
specific for different epitopes. Using an agent-based approach, this model
demonstrates how intercelluar competition can indirectly provide a means
of T cell regulation.
At the cellular level, Figge et al. simulate B cell migration in the germinal
center of a lymph node (Figge et al., 2008). In their model, they assume that
individual B cells move according to a random walk attempting to follow a
chemoattractant. They apply their model for the purpose of resolving the
paradox obtained from two-photon imaging data that B cell migration
initially appears follow a chemotactic gradient but then devolves into
what resembles more of an undirected random walk. Using their simulations, they hypothesize that chemotaxis must remain active throughout the
entire B cell migration process as to maintain a sense of the germinal center.
At the same time, individual B cells downregulate chemokine receptors
causing them to lose sensitivity to the chemical gradient.
On the molecular level, Casal et al. construct an ABM for T cell scanning
of the surface of an APC. In this model, the agents are individual T cell
receptors and peptide sequences that populate the surfaces of interacting
cells (Casal et al., 2005).
The main advantage of agent-based modeling is the ability to account for
probabilistic uncertainty and individual diversity within a large population.
The main difficulty is, on the other hand, the huge computational complexity that accompanies such sophisticated models. Roughly speaking,
most ABMs take on the order of several hours to even days to simulate
even once, whereas most deterministic models can be evaluated much
faster. Furthermore, stochastic ABMs usually have to be simulated numerous times to obtain the overall average behavior of the system. Thus, despite
their advantages ABMs often present great challenges in terms of computational implementation.
2.5. Stochastic differential equations
Perhaps the least explored path in immunological modeling is the use of
stochastic differential equations (SDEs). From the point of view of complexity, SDEs lie somewhere in-between deterministic, differential equation models, and ABMs. SDEs are written and formulated a lot like ODEs,
except that they allow their variables to take random values. Traditionally,
SDEs provide an effective means of accounting for noise, random walks,
Modeling and Simulation of the Immune System
91
and sporadic events (modeled as a Poisson process), and they have been
applied extensively in financial mathematics, chemistry, and physics. However, they have not yet fully made an entry into mainstream immunological
modeling.
Nonetheless, we can provide one example of an SDE model that has
been applied to X-linked agammaglobulinemia, a genetic disorder of B cell
maturation that prevents the production of immunoglobulin. In his model,
Figge formulates a system of SDEs to simulate the depletion of immunoglobulin by natural degradation and antigenic consumption and its periodic
replenishment by immunoglobulin substitution therapy (Figge, 2009). The
stochastic model captures the tendency of the immunoglobulin repertoire
to shift toward certain antigen-specificities at the expense of others.
In addition, the regulatory network clarifies how immunoglobulin substitution therapy may affect other aspects of the overall immune response in
ways that were not clear before. Figge’s assessment of the current treatment
strategy is that lower treatment frequencies, separated by a period of one to
several weeks, may actually benefit the prevention of chronic infection.
The computational complexity of SDEs generally falls between that of
deterministic models and ABMs. SDE models are one step above ODE
models in terms of their complexity, because they incorporate stochastic
effects. Nonetheless, like ODEs, SDEs still consider populations as collective groups rather than as individual agents.
2.6. Which modeling approach is appropriate?
Such a diverse selection of available models, not to mention possible hybrid
formulations, begs the question, ‘‘Which modeling approach is most appropriate?’’ The answer depends on the nature of regulatory interactions
involved, among other issues.
As discussed, ODEs are the most efficient method for modeling huge
levels of biological complexity without a substantial increase in the computational work. For any regulatory networks that do not rely significantly on
delayed feedback, spatial distribution of cells and molecules, or probabilistic
events, ODE models are the most effective approach. For networks that
seem to depend on delayed feedback, DDEs provide a good paradigm and
also remain relatively simple from a computational point of view.
Networks of cells and molecules that do not mix well or efficiently, but
remain localized over a long period of time, may correspond most appropriately to PDE models that account for space. Similarly, networks of cells
that change behavior gradually over time also lend themselves naturally to
age or maturity-structured formulations using PDEs. Moving beyond
deterministic models, SDEs provide one means of adding stochasticity to
differential equations, but they come with a higher level of computational
complexity.
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In recent years, there has been a growth in the use of ABMs. The ABM
paradigm currently provides the most complex and versatile framework for
mathematical modeling by incorporating all elements of spatial and temporal
dynamics, probabilistic events, and individual diversity within populations.
However, ABMs demand by far the most intensive computational algorithms
and are often impractical for statistical analyses such as parameter sensitivity or
data fitting, which usually require numerous simulations. As a paradigm,
ABMs seek to closely replicate the complexity of biological systems, so that
‘‘experiments’’ can be done in silico much more economically and even
ethically than they could be performed in vivo.
On one hand, ABMs provide practical means of transporting experimental studies from the wet lab to the computer lab. However, ABMs do
not replace other forms of modeling because they do not substantially
reduce the inherent complexity of biological systems. Furthermore, when
applying ABMs to an immunological network, one should confirm that the
dynamic behavior of the ABM cannot be sufficiently recreated by a simpler,
differential equation formulation. For example, the two papers (DoumicJauffret, 2009; Kim et al., 2008) succeed in replicating the dynamics of a
highly complex ABM with almost no deviation using two deterministic
models: difference equations and PDEs. In addition, these deterministic
models require only 4 min and 30 s of computation time as opposed to the
approximately 50 7 ¼ 350 h required by the ABM. This result demonstrates the efficacy of hybrid methods that merge both ABM and differential
equation frameworks to capture the underlying characteristics of a
biological network without adding any superfluous detail.
In practice, it is difficult to predict which mathematical and computational paradigm is most suitable for a given situation. Ideally, to thoroughly
understand a system from a modeling perspective, one should devise mathematical models of all types for each immunological network in question.
This line of thinking is, needless to say, unreasonable. Instead, the rational
and the most informed approach to mathematical modeling is to recognize
the capacities and limitations of each type of model and to apply the
paradigm that most accurately quantifies the essential dynamics of the system
without introducing any unnecessary complexity.
3. Two Examples of Models to Understand
T Cell Regulation
In the following section, we provide two examples of DDE models
based on immune regulatory networks that were proposed by Antia et al.
(2003) and Kim (2009). Each of the models describes a distinct network that
could regulate T cell development during an acute infection.
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Modeling and Simulation of the Immune System
3.1. Intracellular regulation: The T cell program
The first regulatory network is based on the notion of a T cell proliferation
program. According to this concept, T cells follow a fixed program of
development that initiates after stimulation and then proceeds to unfold
without any further feedback from the environment. As mentioned in
Section 2.3, a programmed cellular response implies an intracellular regulatory mechanism that may still be highly complex, although it no longer
interacts with the rest of the external network. Our example comes from
Kim (2009), and it stems from the original T cell program model formulated
by Antia et al. (2003) and further developed by Wodarz and Thomsen (2005).
The T cell program (illustrated in Fig. 4.1) can be summarized as follows:
1. APCs mature, present relevant target antigen, and migrate from the site
of infection to the draining lymph node.
2. In the lymph node, APCs activate naı̈ve T cells that enter a minimal
developmental program of m cell divisions.
3. T cells that have completed the minimal developmental program
become effector cells that can divide in an antigen-dependent manner
(i.e., upon further interaction with APCs) up to n additional times.
4. Effector cells that divided the maximum number of times stop dividing.
1) Migration of APCs to lymph node
sA
a(t)A0
A0
A1
2) Initial T cell activation
sT
kA1T0
T0
Delay = s
m
T1 ⫻2
A1
3) Antigen-dependent proliferation
Ti
kA1Ti
Delay = r
Ti + 1 ⫻2
4) Awaiting apoptosis
Tn + 1
A1
(Although not indicated, each cell has a natural death rate according to its kind.)
Figure 4.1 The T cell program. (1) Immature APCs pick up antigen at the site of
infection at a time-dependent rate a(t). These APCs mature and migrate to the lymph
node. (2) Mature antigen-bearing APCs present antigen to naı̈ve T cells causing them to
activate and enter the minimal developmental program of m divisions. (3) Activated
T cells that have completed the minimal program continue to divide upon further
interaction with mature APCs for up to n additional divisions. (4) T cells that have
completed the maximal number of divisions stop dividing and wait for apoptosis.
Although not indicated, each cell in the diagram has a natural death rate according to
its kind.
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Peter S. Kim et al.
This process can be translated into a system of DDEs in which each
equation corresponds to one of the cell populations shown in Fig. 4.1. The
system of DDEs is as follows:
0
A 0 ðtÞ ¼ sA d0 A0 ðtÞ aðtÞA0 ðtÞ;
0
ð4:1Þ
A1 ðtÞ ¼ aðtÞA0 ðtÞ d1 A1 ðtÞ;
ð4:2Þ
T0 ðtÞ ¼ sT d0 T0 ðtÞ kA1 ðtÞT0 ðtÞ;
ð4:3Þ
0
0
T1 ðtÞ ¼ 2m kA1 ðt sÞT0 ðt sÞ kA1 ðtÞT1 ðtÞ d1 T1 ðtÞ;
0
Ti ðtÞ ¼ 2kA1 ðt rÞTi1 ðt rÞ kA1 ðtÞTi ðtÞ d1 Ti ðtÞ;
0
T nþ1 ðtÞ ¼ 2kA1 ðt rÞTn ðt rÞ d1 Tnþ1 ðtÞ:
ð4:4Þ
ð4:5Þ
ð4:6Þ
The variables in the equations have the following definitions:
A0 is the concentration of APCs at the site of infection.
A1 is the concentration of APCs that have matured, started to present
target antigen, and migrated to the lymph node.
T0 is the concentration of antigen-specific naı̈ve T cells in the lymph node.
Tiþ 1 is the concentration of effector cells that undergone i antigendependent divisions after the minimal developmental program.
Tnþ 1, denotes T cells that have undergone n divisions after the minimal
developmental program. These cells have terminated the proliferation
program and can no longer divide.
Cell concentration is measured in units of k/mL (thousands of cells per
microliter).
Figure 4.2 shows an expanded diagram of how the first two equations,
Eqs. (4.1) and (4.2), are derived from step 1. Equation (4.1) pertains to the
1) Migration of APCs to lymph node
sA
Supply
A0
Supply
Stimulation
A0' (t) = sA-d0A0(t) − a(t)A0(t)
Natural death
a(t)A0
Stimulation
Δ
A1
Stimulation
A'1(t) = a(t)A0(t) − d1A1(t)
Natural death
Figure 4.2 Expanded diagram of how Eqs. (4.1) and (4.2) are derived from step 1 of
the T cell program.
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Modeling and Simulation of the Immune System
population of immature APCs waiting at the site of infection. These cells are
supplied into the system at a constant rate, sA, and die at a proportional
rate, d0. Without stimulation, this population always remains at equilibrium,
given by sA/d0. The time-dependent coefficient a(t) denotes the rate of
stimulation of APCs as a function of time. The function a(t) can be seen as
being proportional to the antigen concentration at the site of infection.
Equation (4.2) pertains to the population of APCs that have matured,
started to present relevant antigen, and migrated to the lymph node. For
simplicity, the model accounts for the maturation, presentation of antigen,
and migration of APCs as one event. The first term of the equation
corresponds to the rate at which these APCs enter the lymph node as
APCs at the site of infection are stimulated. The second term is the natural
death rate of this population.
Figure 4.3 shows an expanded diagram of how Eqs. (4.3) and (4.4) are
derived from step 2. Equation (4.3) pertains to naı̈ve T cells. This population is replenished at a constant rate, sT, and dies at a proportional rate, d0.
Without stimulation, the population remains at equilibrium, sT/d0. The
third term in this equation is the rate of stimulation of naı̈ve T cells by
mature APCs. The bilinear form of this term follows the law of mass action
where k is the kinetic coefficient.
Equation (4.4) pertains to newly differentiated effector cells that have
just finished the minimal developmental program of m divisions. The first
term gives the rate at which activated naı̈ve T cells enter the first effector
state, T1. This term corresponds to the final term of the previous equation
for T00 (t), except that it has an additional coefficient of 2m and it depends on
cell concentrations at time ts. The coefficient 2m accounts for the increase
in population of naı̈ve T cells after m divisions, and the time delay s is the
2) Initial T cell activation
Stimulation
ST
Supply
T0
Proliferation from
minimal developmental program
kA1T0
m
T1 × 2
Delay = s
A1
Supply
Stimulation
T0'(t) = sT − d1T0(t) − kA1(t)T0(t)
Natural death
Proliferation from
minimal developmental program
Natural death
T1⬘(t) = 2mkA1(t − s)T0(t − s) − kA1(t)T1(t) − d1T1(t)
Further stimulation
Figure 4.3 Expanded diagram of how Eqs. (4.3) and (4.4) are derived from step 2 of
the T cell program.
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Peter S. Kim et al.
duration of the minimal developmental program. This term accounts for
newly proliferated effector cells that appear in the T1 population s time
units after activation from T0. The second term is the rate at which T1 cells
are stimulated by mature APCs for further division. It is based on the law of
mass action and is of the same form as the final term of the equation for
T00 (t). This term exists in the equation only if the number of possible
antigen-dependent divisions, n, is not 0. Finally, as shown by the last
term, T1 cells continuously die at rate d1.
Figure 4.4 shows an expanded diagram of how Eq. (4.5) is derived from
step 3 and how Eq. (4.6) is derived from step 4. For i ¼ 2, ..., n, Eq. (4.5) for
Ti0 (t) is analogous to the equation for T10 (t), except that these cells only
divide once after stimulation. Hence, the coefficient of the first term is 2,
and the time delay is r, the duration of a single division. As before, the
second term is the rate at which these cells become stimulated for further
division, and the final term is the death rate. Note that we use the same
death rate, d1, for all effector cells.
The final equation, Eq. (4.6), pertains to cells that have undergone the
maximum number of possible antigen-dependent divisions. These cells do
not divide anymore and can only die at rate d1.
The parameter estimates used for this model come from Kim (2009) and
are summarized in Table 4.2. The function a(t), representing the rate of
antigen stimulation, is defined by
fðtÞfðb tÞ
aðtÞ ¼ c
;
ð4:7Þ
fðbÞ2
where
fðxÞ ¼
e1=x
0
2
3) Antigen-dependent proliferation
Ti
Δ
kA1Ti
Delay = r
4) Awaiting apoptosis
Ti+1 ×2
Tn⫹1
(after n divisions)
A1
Proliferation
if x 0
if x < 0
Further stimulation
Ti'(t) = 2kA1(t − r)Ti-1(t − r) − kA1(t)Ti (t) − d1Ti(t)
Natural death
Proliferation
T 'n+1(t) = 2kA1(t − ρ)Tn(t − r) − d1Tn+1(t)
Natural death
Figure 4.4 Expanded diagram of how Eq. (4.5) is derived from step 3 and how (4.6) is
derived from step 4 of the T cell program.
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Modeling and Simulation of the Immune System
Table 4.2 Estimates for model parameters
Parameter
Description
Estimate
A0(0)
T0(0)
sA
sT
d0
d0
d1
d1
k
m
Initial concentration of immature APCs
Initial concentration of naı̈ve T cells
Supply rate of immature APCs
Supply rate of naı̈ve T cells
Death/turnover rate of immature APCs
Death/turnover rate of naı̈ve T cells
Death/turnover rate of mature APCs
Death/turnover rate of effector T cells
Kinetic coefficient
Number of divisions in minimal
developmental program
Maximum number of antigendependent divisions
Duration of one T cell division
Duration of minimal developmental
program
Rate of APC stimulation
Duration of antigen availability
Level of APC stimulation
Rate of differentiation of effector cells
into iTregs
sA/d0 ¼ 10
sT/d0 ¼ 0.04
0.3
0.0012
0.03
0.03
0.8
0.4
20
7
n
r
s
a(t)
b
c
r
3–10
1/3
3
Eq. (4.7)
10
1
0.01
Concentrations are in units of k/mL, and time is measured in days.
and b, c > 0. This function starts at 0, increases to a positive value for some
time, and returns to 0. Kim (2009) demonstrated the duration of antigen
availability, b, is estimated to be 10 days, and the level of APC stimulation, c,
is estimated to be 1. (See Fig. 4.5 for graphs of a(t) for b ¼ 3 and b ¼ 10
when c ¼ 1.)
3.2. Intercellular regulation: iTreg-based negative feedback
The second regulatory network is based on a negative feedback loop
mediated by iTregs that differentiate from effector T cells during the course
of the immune response. Several mathematical models considering Tregmediated feedback have been developed for naturally occurring regulatory
T cells (nTregs) (Burroughs et al., 2006; León et al., 2003) and iTregs
(Fouchet and Regoes, 2008), but these models focus on the function of
Tregs in maintaining immune tolerance. In contrast, the following model
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Peter S. Kim et al.
focuses on the primary response against acute infection rather than longterm behavior. The model comes from Kim’s study (2009).
In this feedback network, T cell responses begin the same way as for the
T cell program. However, T cell contraction initiates differently, since it is
mediated by external suppression by iTregs. This process (illustrated in
Fig. 4.6) can be described in five steps:
1. APCs mature, present relevant target antigen, and migrate from the site
of infection to the draining lymph node.
a (t)
1
b=3
b = 10
0.5
0
0
2
4
6
8
10
t (days)
Figure 4.5 Graphs of the antigen function a(t) given by Eq. (4.7) for b ¼ 3 and b ¼ 10
when c ¼ 1. The function a(t) represents the rate that immature APCs pick up antigen
and are stimulated.
1) Migration of APCs to lymph node
SA
a(t)A0
Δ
A0
A1
2) Initial T cell activation
ST
kA1T0
T0
Delay = s
Δ
TE ×2m
A1
3) Antigen-dependent proliferation
TE
kA1TE
Delay = r
4) Effector cells differentiate into iTregs
TE ⫻2
TE
rTE
TR
A1
5) iTregs suppress effector cells
TR
TE
kTRTE
T1
(although not indicated, each cell has a natural death rate according to its kind.)
Figure 4.6 Diagram of the iTreg model. The first three steps are identical to those in
the cell division-based model that is shown in Fig. 4.1. In the fourth step, effector cells
differentiate into iTregs at rate r. In the fifth step, iTregs suppress effector cells.
Although not indicated, each cell in the diagram has a natural death rate according to
its kind.
Modeling and Simulation of the Immune System
99
2. In the lymph node, APCs activate naı̈ve T cells that enter a minimal
developmental program of m cell divisions.
3. T cells that have completed the minimal developmental program
become effector cells that keep dividing in an antigen-dependent manner as long as they are not suppressed by iTregs.
4. Effector cells differentiate into iTregs at a constant rate.
5. The iTregs suppress effector cells upon interaction.
The model can be formulated as a system of five DDEs shown below:
A0
0 ðtÞ
¼ sA d0 A0 ðtÞ aðtÞA0 ðtÞ;
A 1 ðtÞ ¼ aðtÞA0 ðtÞ d1 A1 ðtÞ;
0
T 0 ðtÞ ¼ sT d0 T0 ðtÞ kA1 ðtÞT0 ðtÞ;
0
T E ðtÞ ¼ 2m kA1 ðt sÞT0 ðt sÞT0 ðt sÞ
kA1 ðtÞTE ðtÞ þ 2kA1 ðt rÞTE ðt rÞ
ðd1 þ rÞTE ðtÞ kTR ðtÞTE ðtÞ:
0
0
T R ðtÞ ¼ rTE ðtÞ d1 TR :
ð4:8Þ
ð4:9Þ
As in the previous model, A0 is the concentration of APCs at the site of
infection, A1 is the concentration of APCs that have matured, started to
present target antigen, and migrated to the lymph node, and T0 is the
concentration of naı̈ve T cells in the lymph node. In addition, TE is the
concentration of effector cells, and TR is the concentration of iTregs.
The first three equations for APCs and naı̈ve T cells are identical to those
in the T cell program model from Section 3.1. The first two terms of
Eq. (4.8) for TE0 (t) are identical to the first two terms of Eq. (4.4) for the
T cell program. The third term in this equation is the rate that cells that have
just finished dividing reenter the effector cell population. In this model, cells
do not have a programmed maximum number of divisions, so it is not
necessary to count the number of divisions a cell has undertaken. The only
regulatory mechanism is suppression by iTregs. The fourth term is the rate
that effector cells exit the population through death at rate d1 or differentiate
into iTregs at rate r. The final term is the rate that effector cells are
suppressed by iTregs. As before, the rate of iTreg–effector interactions
follows the same mass action law as APC–T cell interactions.
Equation (4.9) pertains to iTregs. The first term is the rate at which
effector cells differentiate into iTregs, and the second term is the rate at
which iTregs die. The iTregs have the same death rate as effector cells.
All parameters in this model are identical to those used for the T cell
program, except for r, the rate of differentiation of effector cells into iTregs.
As Kim (2009) demonstrated, we estimate that r ¼ 0.01/day, meaning that
1% of effector cells differentiate into iTregs per day. The parameters used in
the iTreg model are listed in Table 4.2.
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Peter S. Kim et al.
4. How to Implement Mathematical Models in
Computer Simulations
Once a mathematical model has been developed, the next step is to
implement it computationally. A common approach is to write the relevant
computational software for each problem, since this method has the advantage of allowing the programmer to optimize the computer algorithms for
his or her particular needs. However, various software packages already exist
for most of the modeling paradigms. For ABM simulations, the immune
system simulator (IMMSIM) (Celada and Seiden, 1992; Seiden and Celada,
1992), the synthetic immune system (SIS) (Mata and Cohn, 2007), the basic
immune simulator (BIS) (Folcik et al., 2007) provide platforms for generating virtual immune systems populated by a variety of cell types. For
deterministic, differential equation models, the most frequently used programs are MATLAB, Maple, and XPPAUT. In general, DDE models are
relatively simple to evaluate on any of the software platforms for differential
equations mentioned above, and we numerically simulate the DDE models
from Sections 3.1 and 3.2 with the ‘‘dde23’’ function of MATLAB R2008a.
Currently, no widely used computational tools exist for evaluating SDE
models for biological systems, since this direction of research has yet to be
developed. Although numerous programs are available for pricing financial
devices, these approaches are usually not ideal for models pertaining to
immunological networks.
4.1. Simulation of the T cell program
The following simulations are drawn from Kim’s study (2009) and primarily
focus on the effect of antigen stimulation levels and precursor concentrations on the magnitude of the T cell response. Before proceeding to
numerical simulations, the first crucial point to notice is that the T cell
dynamics given by Eqs. (4.1)–(4.6) directly scale with respect to the precursor concentration, T0(0). In other words, a T cell response that begins with
x times as many precursors as another automatically has a peak that is x times
higher. This scaling property holds, because T cell program model is linear
with respect to the T cell populations, Ti(0). As a result, simulations
pertaining to the T cell program only consider relative T cell expansion
levels, given by Ttotal(t)/T0(0), rather than total T cell populations given by
ðt
n X
Ttotal ðtÞ ¼
Ti ðtÞ þ
kA1 ðuÞdu þ Tnþ1 ðtÞ:
i¼1
tr
Note that Eqs. (4.4)–(4.6) imply that stimulated T cells leave the system
during the division process and return r time units later. Hence, the total T
cell concentration is not only the sum of T cell populations given by Ti(t),
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Modeling and Simulation of the Immune System
but also the populations that are undergoing division, which are given by
the integrals in the above expression.
The first set of simulations examines the dependence of the T cell peak
on the two antigen-related parameters, c and b, corresponding to the level
and duration of antigen presentation, respectively. We use the parameters
listed in Table 4.2, and c and b vary from 0.1 to 3 and from 1 to 15 days,
respectively. The maximum T cell expansion level versus c is plotted in
Fig. 4.7A, and the maximum T cell expansion level versus b is plotted in
4.7B. To understand the T cell behavior under a variety of possible T cell
programs, we use the lowest and highest estimated values (3 and 10) of n,
which denotes the maximum possible number of antigen-dependent divisions after the minimal developmental program. We draw the two
corresponding curves for each value of n in each plot.
As can be seen in Fig. 4.7A, T cell dynamics saturate very quickly in
relation to c, so much so that the doubling level period is almost constant
A
Population doublings
25
20
n = 10
15
n=3
10
5
0
0
0.5
1
B
1.5
c
2
2.5
3
Population doublings
25
20
n = 10
15
n=3
10
5
0
0
5
10
15
b
Figure 4.7 Dependence of T cell dynamics on c, a parameter corresponding to the level
of antigen presentation, and on b, the duration of antigen availability. (A) Maximum
T cell expansion level versus c. Expansion level is measured in population doublings,
which is defined by log2(max(Ttotal/T0(0))). Data is shown for the two possible values of
n, the maximum number of possible antigen-dependent divisions after the minimal
developmental program. (B) Maximum T cell expansion level versus b.
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Peter S. Kim et al.
from as low as c ¼ 0.1 to as high as c ¼ 3. By continuity, the size of the
T cell peak must go to 0 as c decreases, but the drop is very steep. The two
extra points shown in the curves for n ¼ 3 correspond to c ¼ 0.001 and
c ¼ 0.01. These values correspond to roughly 0.1% and 1% of APCs getting
stimulated per day. Hence, even very low stimulation levels result in nearly
saturated T cell dynamics.
The plots of the maximum expansion level and the time of peak versus
b are shown in Fig. 4.7B. The figure shows that T cell dynamics also saturate
as b increases, but not as quickly as for c. Hence, the simulations show that
the duration of antigen availability is more important than the level. The
plots on Fig. 4.7B show that for both n ¼ 3 and n ¼ 10, T cell expansion
levels begin to saturate around b ¼ 4 or 5 days, indicating that the immune
response behaves similarly as long as antigen remains available for long
enough to elicit a fully developed T cell response.
As a final simulation for the T cell program model, Fig. 4.8 shows the
time evolution of various cell populations when n ¼ 10 and the rest of the
A
4000
Effector cells
Concentration (k/mL)
3500
3000
2500
2000
1500
1000
500
0
Concentration (k/mL)
B
10,000 ⫻ naive cells
0
10
9
8
7
6
5
4
3
2
1
0
5
10
Time (day)
15
20
Immature APCs
Mature APCs
0
5
10
Time (day)
15
20
Figure 4.8 Time evolution of immune cell populations over time. (A) The dynamics of
naı̈ve and effector cells over 20 days. (B) The dynamics of immature and mature APCs.
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Modeling and Simulation of the Immune System
parameters are taken from Table 4.2. In Fig. 4.8A, the T cell peak is 94605
times higher than the precursor concentration, T0(0) ¼ 0.04 k/mL.
As mentioned at the beginning of this section, the ratio between the T
cell peak and the precursor concentration remains constant for all values of
T0(0). Since this relation is exactly linear, we do not give a plot of the
maximum height of T cell expansion versus precursor concentration.
4.2. Simulation of the iTreg model
The simulations in this section are also drawn from Kim’s study (2009) and
follow a similar pattern to those in Section 4.1. Due to the negative
feedback from iTregs in Eq. (4.8), T cell dynamics do not directly scale
with respect to precursor frequencies as in the T cell program model. In this
case, it is informative to look at the total effector cell population, given by
Ttotal ðtÞ ¼ T1 ðtÞ þ
ðt
kA1 ðuÞT1 ðuÞdu:
tr
Figure 4.9 displays a log–log plot of the maximum expansion level versus
the initial naı̈ve T cell concentration, T0(0), which is varied from 4 10 4
to 4 k/mL, a range 100 times lower and 100 times higher than the estimated
value in Table 4.2. As shown in Fig.4.9A, the simulated data fits a power law
of exponent 0.3004, meaning that the expansion roughly scales to the cubed
3.5
log10 max(Ttotal)
p1 = 1327
p2 = 0.3004
(rcorr = 0.9987)
3
max(Ttotal) = p1T0(0)p2
2.5
2
−3.5
−3
−2.5
−2
−1.5 −1 −0.5
log10 T0(0)
0
0.5
1
Figure 4.9 A log–log plot of the dependence of the T peak on T0(0), the initial
concentration of naı̈ve T cells. The linear regression shows that the maximum T cell
expansion level is roughly proportional to T0(0)1/3. The linear correlation
rcorr ¼ 0.9987.
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Peter S. Kim et al.
root of the initial naı̈ve cell concentration. For example, to obtain a T cell
response that is 10 times higher (or lower) than normal, the system would
need to start with a reactive precursor concentration that is 1000 times
higher (or lower) than normal.
Following the same sensitivity analysis as in Section 3.1, we see in
Fig. 4.10 that the dynamics of the iTreg model exhibits similar saturating
behavior with respect to the level and to the duration of antigen stimulation,
given by c and b, respectively. Like the program-based model, the feedback
model generates dynamics that behave insensitively to the level and duration
of antigen stimulation.
Figure 4.11 shows the time evolution of the effector and iTreg populations when all other parameters are taken from Table 4.2. The figure
A
16
Population doublings
14
12
10
8
6
4
2
0
0
0.5
1
1.5
c
B
2
2.5
3
14
Population doublings
12
10
8
6
4
2
0
0
5
10
b
Figure 4.10 Dependence of T cell dynamics on c, a parameter corresponding to the
level of antigen presentation, and on b, the duration of antigen availability. (A) Maximum T cell expansion level versus c. (B) Maximum T cell expansion level versus b.
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Modeling and Simulation of the Immune System
500
450
Concentration (k/mL)
400
Total effector cells
350
300
250
200
150
100
100 ⫻ iTregs
50
0
0
5
10
Time (day)
15
20
Figure 4.11 Time evolution of effector and iTreg populations over time. The peak of
the iTreg response roughly coincides with the peak of the T cell response, but the iTreg
response decays slower.
indicates that the iTreg concentration peaks around the same time as the
T cell response, but lingers a while longer ensuring a full contraction of
the T cell population. In this example, the naı̈ve T cell population begins at
0.04 k/mL and peaks at 482 k/mL, corresponding to an expansion level of
13.6 divisions on average.
The numerical simulations show that both the T cell program and
feedback regulation models exhibit similar insensitivity to the nature of
antigen stimulation. However, the feedback model behaves differently
from the program-based model with respect to variations in precursor
frequency. Specifically, the feedback model significantly reduces variance
in precursor concentration (by a cubed root power law), whereas the T cell
program model directly translates variance in precursor concentration to
variance in peak T cell levels (by a linear scaling law).
5. Concluding Remarks
By constructing mathematical models based on DDEs, we show how
we can investigate two structurally distinct, regulatory networks for T cell
dynamics. Using modeling, we readily determine the similarities and differences between the two models. In particular, we find that both networks
have very low sensitivity to changes in the nature of antigen stimulation, but
differ greatly in how they respond to variations in T cell precursor
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Peter S. Kim et al.
frequency. Hence, our example demonstrates how mathematical and
computational analysis can immediately provide a testable hypothesis to
help validate or invalidate these two proposed regulatory networks.
Moving to a broader perspective, the entire immune response operates
as a system of self-regulating networks, and many of these networks have the
potential to be elucidated by mathematical modeling and computational
simulation. Several modeling frameworks already exist and up to now,
ODE models have been the most widely used due to their versatility across
a wide range of problems and their ability to handle complex systems
efficiently. DDEs and PDEs, which are both infinite-dimensional systems,
also frequently appear in the repertoire of deterministic models. DDEs
possess one advantage over ODEs in that they explicitly account for the
delayed feedback without adding substantial computational complexity.
PDE models provide an even more complex framework and can incorporate a wide range of spatial and temporal phenomena such as molecular
diffusion and cell motion and maturation.
Among probabilistic models, stochastic ABMs are the most widely used,
since they are typically easy to formulate, directly model individual diversity
within populations, and recreate phenomena resulting from random events.
An unavoidable disadvantage of ABMs is, however, that they are computationally demanding, especially in comparison to deterministic, differential
equation models that can often be used to approximate the same phenomena with good accuracy. Thus, a promising compromise between the
deterministic, differential equation, and stochastic agent-based paradigms
comes from SDEs, a type of differential equation system that incorporates
stochastic behavior. Nonetheless, this domain of mathematical modeling
remains largely unexplored, at least for immunological networks, and offers
a strong possibility for future research.
The immune regulatory network is intricate, made up of myriad intracellular and intercellular interactions that evade complete understanding and
provide fertile ground for the unraveling of these interwoven mysteries with
the help of insight gained from mathematical and computational modeling.
ACKNOWLEDGMENTS
The work of PSK was supported in part by the NSF Research Training Grant and the
Department of Mathematics at the University of Utah. The work of DL was supported in
part by the joint NSF/NIGMS program under Grant Number DMS-0758374. This work
was supported by a Department of Defense Era of Hope grant to PPL. The work of DL and
of PPL was supported in part by Grant Number R01CA130817 from the National Cancer
Institute. The content is solely the responsibility of the authors and is does not necessarily
represent the official views of the National Cancer Institute or the National Institute of
Health.
Modeling and Simulation of the Immune System
107
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